Brauer Group

Abstract

This chapter continues the study of finite-dimensional associative division algebras over a field F, with particular attention to those that are simple and have center F. Section 5 is a selfcontained digression on cohomology of groups that is preparation for an application in Section 6 and for a general treatment of homological algebra in Chapter IV.

Section 1 introduces the Brauer group of F and the relative Brauer group of K/F, K being any finite extension field. The Brauer group B(F) is the abelian group of equivalence classes of finite-dimensional central simple algebras over F under a relation called Brauer equivalence. The inclusion F ⊆ K induces a group homomorphism B(F) → B(K), and the relative Brauer group B(K/F) is the kernel of this homomorphism. The members of the kernel are those classes such that the tensor product with K of any member of the class is isomorphic to some full matrix algebra Mn(K); such a class always has a representative A with dimf A = (dimf K)2. One proves that B(F) is the union of all B(K/F) as K ranges over all finite Galois extensions of F.

Sections 2–3 establish a group isomorphism B(K/F)≌ H2(Gal(K/F), Kx) when K is a finite Galois extension of F. With these hypotheses on K and F, Section 2 introduces data called a factor set for each member of B(K/F). The data depend on some choices, and the effect of making different choices is to multiply the factor set by a “trivial factor set.” Passage to factor sets thereby yields a function from B(K/F) to the cohomology group H2(Gal(K/F), Kx). Section 3 shows how to construct a concrete central simple algebra over F from a factor set, and this construction is used to show that the function from B(K/F) to H2(Gal(K/F), Kx) constructed in Section 2 is one-one onto. An additional argument shows that this function in fact is a group isomorphism.

Section 4 proves under the same hypotheses that H1(Gal(K/F), Kx) = 0, and a corollary makes this result concrete when the Galois group is cyclic. This result and the corollary are known as Hilbert’s Theorem 90.

Section 5 is a self-contained digression on the cohomology of groups. If G is a group and ZG is its integral group ring, a standard resolution of Z by free ZG modules is constructed in the category of all unital left ZG modules. This has the property that if M is an abelian group on which G acts by automorphisms, then the groups Hn(G, M) result from applying the functor HomZG(., M) to the members of this resolution, dropping the term HomZG(Z, M), and taking the cohomology of the resulting complex. Section 5 goes on to show that the groups Hn(G, M) arise whenever this construction is applied to any free resolution of Z, not necessarily the standard one. This section serves as a prerequisite for Section 6 and as motivational background for Chapter IV.

Section 6 applies the result of Section 5 in the case thatG is finite cyclic, producing a nonstandard free resolution of Z in this case. From this alternative free resolution, one obtains a rather explicit formula for H2(G, M) whenever G is finite cyclic. Application to the case that G is the Galois group Gal(K/F) for a finite Galois extension gives the explicit formula B(K/F)≌Fx/NK/F (Kx) for the relative Brauer group when the Galois group is cyclic.